Why isn't there any command to start a paragraph with \vbox?

The command for starting horizontal mode with a \vbox is \leavevmode. Why would you need something else?

Your assumption is wrong, I'm afraid: it's not “empty contents” that triggers mode change. It is precisely \unhbox that does.

Differently from \box and \copy, \unhbox is a horizontal command, so it triggers horizontal mode if TeX is in vertical mode. The macro \leavevmode uses \voidb@x so that nothing is appended to the horizontal list (after possibly having started it).

The analog command \unvbox is a vertical command and ends a paragraph.

\catcode`@=11
\def\leavehmode{\unvbox\voidb@x}
\catcode`@=12

\everypar{PAR}% to show a paragraph has started

abc\leavehmode def

\bye

I don't know whether this is clear or not, but let's state it explicitly (from the TeXbook):

|\leavevmode| is an abbreviation for `|\unhbox\voidbox|',
where |\voidbox| is a permanently empty box register.

Now what does \unhbox\voidbox actually do? Let's take a look in tex.web:

@ @<Cases of |main_control| that build...@>=
vmode+start_par: new_graf(cur_chr>0);
vmode+letter,vmode+other_char,vmode+char_num,vmode+char_given,
   vmode+math_shift,vmode+un_hbox,vmode+vrule,
   vmode+accent,vmode+discretionary,vmode+hskip,vmode+valign,
   vmode+ex_space,vmode+no_boundary:@t@>@;@/
  begin back_input; new_graf(true);
  end;

As you can see any of those combinations of mode and something else start a paragraph, but vmode+un_vbox is not among them.

On a personal note, I find it interesting that Knuth did not let vmode+<*>vbox start a paragraph but vmode+valign does.